The Calabi Problem for Fano Threefolds

This book determines whether the general element of each family of Fano threefolds is K-polystable, a major problem in mathematics.

Carolina Araujo (Author), Ana-Maria Castravet (Author), Ivan Cheltsov (Author), Kento Fujita (Author), Anne-Sophie Kaloghiros (Author), Jesus Martinez-Garcia (Author), Constantin Shramov (Author), Hendrik Süß (Author), Nivedita Viswanathan (Author)

9781009193399, Cambridge University Press

Paperback / softback, published 29 June 2023

455 pages
22.9 x 15.2 x 2.5 cm, 0.66 kg

'It is a difficult problem to check whether a given Fano variety is K-polystable. This book settles this problem for the general members of all the 105 deformation families of smooth Fano 3-folds. The book is recommended to anyone interested in K-stability and existence of Kähler-Einstein metrics on Fano varieties.' Caucher Birkar FRS, Tsinghua University and University of Cambridge

Algebraic varieties are shapes defined by polynomial equations. Smooth Fano threefolds are a fundamental subclass that can be thought of as higher-dimensional generalizations of ordinary spheres. They belong to 105 irreducible deformation families. This book determines whether the general element of each family admits a Kähler–Einstein metric (and for many families, for all elements), addressing a question going back to Calabi 70 years ago. The book's solution exploits the relation between these metrics and the algebraic notion of K-stability. Moreover, the book presents many different techniques to prove the existence of a Kähler–Einstein metric, containing many additional relevant results such as the classification of all Kähler–Einstein smooth Fano threefolds with infinite automorphism groups and computations of delta-invariants of all smooth del Pezzo surfaces. This book will be essential reading for researchers and graduate students working on algebraic geometry and complex geometry.

Introduction
1. K-stability
2. Warm-up: smooth del Pezzo surfaces
3. Proof of main theorem: known cases
4. Proof of main theorem: special cases
5. Proof of main theorem: remaining cases
6. The big table
7. Conclusion
Appendix. Technical results used in proof of main theorem
References
Index.

Subject Areas: Geometry [PBM]